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Derivation

1. Split input into 3 regimes

2. if k < 1.4114890122776786e+29

   1. Initial program 49.8

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]

   2. Initial simplification33.8

      \[\leadsto \frac{\frac{\frac{2}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}{\sin k \cdot \tan k}}{\frac{k}{t} \cdot \frac{k}{t}}\]
   3. Using strategy rm

   4. Applied times-frac33.6

      \[\leadsto \frac{\color{blue}{\frac{\frac{2}{t}}{\sin k} \cdot \frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}}{\frac{k}{t} \cdot \frac{k}{t}}\]

   5. Applied times-frac22.3

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{t}}{\sin k}}{\frac{k}{t}} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{\frac{k}{t}}}\]

   6. Simplified21.3

      \[\leadsto \color{blue}{\frac{\frac{2}{k}}{\sin k}} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{\frac{k}{t}}\]
   7. Using strategy rm

   8. Applied *-un-lft-identity21.3

      \[\leadsto \frac{\frac{2}{k}}{\sin k} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{\color{blue}{1 \cdot \frac{k}{t}}}\]

   9. Applied *-un-lft-identity21.3

      \[\leadsto \frac{\frac{2}{k}}{\sin k} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\color{blue}{1 \cdot \tan k}}}{1 \cdot \frac{k}{t}}\]

   10. Applied times-frac20.3

       \[\leadsto \frac{\frac{2}{k}}{\sin k} \cdot \frac{\color{blue}{\frac{\frac{\ell}{t}}{1} \cdot \frac{\frac{\ell}{t}}{\tan k}}}{1 \cdot \frac{k}{t}}\]

   11. Applied times-frac13.8

       \[\leadsto \frac{\frac{2}{k}}{\sin k} \cdot \color{blue}{\left(\frac{\frac{\frac{\ell}{t}}{1}}{1} \cdot \frac{\frac{\frac{\ell}{t}}{\tan k}}{\frac{k}{t}}\right)}\]

   12. Applied associate-*r*12.3

       \[\leadsto \color{blue}{\left(\frac{\frac{2}{k}}{\sin k} \cdot \frac{\frac{\frac{\ell}{t}}{1}}{1}\right) \cdot \frac{\frac{\frac{\ell}{t}}{\tan k}}{\frac{k}{t}}}\]

   13. Simplified8.5

       \[\leadsto \left(\frac{\frac{2}{k}}{\sin k} \cdot \frac{\frac{\frac{\ell}{t}}{1}}{1}\right) \cdot \color{blue}{\frac{\frac{\ell}{k}}{\tan k}}\]
   14. Using strategy rm

   15. Applied pow18.5

       \[\leadsto \left(\frac{\frac{2}{k}}{\sin k} \cdot \frac{\frac{\frac{\ell}{t}}{1}}{1}\right) \cdot \color{blue}{{\left(\frac{\frac{\ell}{k}}{\tan k}\right)}^{1}}\]

   16. Applied pow18.5

       \[\leadsto \left(\frac{\frac{2}{k}}{\sin k} \cdot \color{blue}{{\left(\frac{\frac{\frac{\ell}{t}}{1}}{1}\right)}^{1}}\right) \cdot {\left(\frac{\frac{\ell}{k}}{\tan k}\right)}^{1}\]

   17. Applied pow18.5

       \[\leadsto \left(\color{blue}{{\left(\frac{\frac{2}{k}}{\sin k}\right)}^{1}} \cdot {\left(\frac{\frac{\frac{\ell}{t}}{1}}{1}\right)}^{1}\right) \cdot {\left(\frac{\frac{\ell}{k}}{\tan k}\right)}^{1}\]

   18. Applied pow-prod-down8.5

       \[\leadsto \color{blue}{{\left(\frac{\frac{2}{k}}{\sin k} \cdot \frac{\frac{\frac{\ell}{t}}{1}}{1}\right)}^{1}} \cdot {\left(\frac{\frac{\ell}{k}}{\tan k}\right)}^{1}\]

   19. Applied pow-prod-down8.5

       \[\leadsto \color{blue}{{\left(\left(\frac{\frac{2}{k}}{\sin k} \cdot \frac{\frac{\frac{\ell}{t}}{1}}{1}\right) \cdot \frac{\frac{\ell}{k}}{\tan k}\right)}^{1}}\]

   20. Simplified7.6

       \[\leadsto {\color{blue}{\left(\frac{\frac{2}{\sin k} \cdot \frac{\frac{\ell}{t}}{k}}{\frac{\tan k}{\frac{\ell}{k}}}\right)}}^{1}\]

   if 1.4114890122776786e+29 < k < 2.3301504039808588e+240

   1. Initial program 45.3

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]

   2. Initial simplification25.7

      \[\leadsto \frac{\frac{\frac{2}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}{\sin k \cdot \tan k}}{\frac{k}{t} \cdot \frac{k}{t}}\]
   3. Using strategy rm

   4. Applied times-frac25.7

      \[\leadsto \frac{\color{blue}{\frac{\frac{2}{t}}{\sin k} \cdot \frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}}{\frac{k}{t} \cdot \frac{k}{t}}\]

   5. Applied times-frac15.1

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{t}}{\sin k}}{\frac{k}{t}} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{\frac{k}{t}}}\]

   6. Simplified15.1

      \[\leadsto \color{blue}{\frac{\frac{2}{k}}{\sin k}} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{\frac{k}{t}}\]
   7. Using strategy rm

   8. Applied *-un-lft-identity15.1

      \[\leadsto \frac{\frac{2}{k}}{\sin k} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{\color{blue}{1 \cdot \frac{k}{t}}}\]

   9. Applied *-un-lft-identity15.1

      \[\leadsto \frac{\frac{2}{k}}{\sin k} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\color{blue}{1 \cdot \tan k}}}{1 \cdot \frac{k}{t}}\]

   10. Applied times-frac15.0

       \[\leadsto \frac{\frac{2}{k}}{\sin k} \cdot \frac{\color{blue}{\frac{\frac{\ell}{t}}{1} \cdot \frac{\frac{\ell}{t}}{\tan k}}}{1 \cdot \frac{k}{t}}\]

   11. Applied times-frac10.2

       \[\leadsto \frac{\frac{2}{k}}{\sin k} \cdot \color{blue}{\left(\frac{\frac{\frac{\ell}{t}}{1}}{1} \cdot \frac{\frac{\frac{\ell}{t}}{\tan k}}{\frac{k}{t}}\right)}\]

   12. Applied associate-*r*8.5

       \[\leadsto \color{blue}{\left(\frac{\frac{2}{k}}{\sin k} \cdot \frac{\frac{\frac{\ell}{t}}{1}}{1}\right) \cdot \frac{\frac{\frac{\ell}{t}}{\tan k}}{\frac{k}{t}}}\]

   13. Simplified3.6

       \[\leadsto \left(\frac{\frac{2}{k}}{\sin k} \cdot \frac{\frac{\frac{\ell}{t}}{1}}{1}\right) \cdot \color{blue}{\frac{\frac{\ell}{k}}{\tan k}}\]
   14. Using strategy rm

   15. Applied frac-times3.6

       \[\leadsto \color{blue}{\frac{\frac{2}{k} \cdot \frac{\frac{\ell}{t}}{1}}{\sin k \cdot 1}} \cdot \frac{\frac{\ell}{k}}{\tan k}\]

   16. Applied frac-times3.6

       \[\leadsto \color{blue}{\frac{\left(\frac{2}{k} \cdot \frac{\frac{\ell}{t}}{1}\right) \cdot \frac{\ell}{k}}{\left(\sin k \cdot 1\right) \cdot \tan k}}\]

   17. Simplified6.1

       \[\leadsto \frac{\color{blue}{\frac{\frac{\ell}{k}}{\frac{k}{\ell}} \cdot \frac{2}{t}}}{\left(\sin k \cdot 1\right) \cdot \tan k}\]

   18. Simplified6.1

       \[\leadsto \frac{\frac{\frac{\ell}{k}}{\frac{k}{\ell}} \cdot \frac{2}{t}}{\color{blue}{\tan k \cdot \sin k}}\]

   if 2.3301504039808588e+240 < k

   1. Initial program 34.6

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]

   2. Initial simplification22.2

      \[\leadsto \frac{\frac{\frac{2}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}{\sin k \cdot \tan k}}{\frac{k}{t} \cdot \frac{k}{t}}\]
   3. Using strategy rm

   4. Applied times-frac22.2

      \[\leadsto \frac{\color{blue}{\frac{\frac{2}{t}}{\sin k} \cdot \frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}}{\frac{k}{t} \cdot \frac{k}{t}}\]

   5. Applied times-frac16.1

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{t}}{\sin k}}{\frac{k}{t}} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{\frac{k}{t}}}\]

   6. Simplified15.9

      \[\leadsto \color{blue}{\frac{\frac{2}{k}}{\sin k}} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{\frac{k}{t}}\]
   7. Using strategy rm

   8. Applied *-un-lft-identity15.9

      \[\leadsto \frac{\frac{2}{k}}{\sin k} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{\color{blue}{1 \cdot \frac{k}{t}}}\]

   9. Applied *-un-lft-identity15.9

      \[\leadsto \frac{\frac{2}{k}}{\sin k} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\color{blue}{1 \cdot \tan k}}}{1 \cdot \frac{k}{t}}\]

   10. Applied times-frac16.0

       \[\leadsto \frac{\frac{2}{k}}{\sin k} \cdot \frac{\color{blue}{\frac{\frac{\ell}{t}}{1} \cdot \frac{\frac{\ell}{t}}{\tan k}}}{1 \cdot \frac{k}{t}}\]

   11. Applied times-frac11.3

       \[\leadsto \frac{\frac{2}{k}}{\sin k} \cdot \color{blue}{\left(\frac{\frac{\frac{\ell}{t}}{1}}{1} \cdot \frac{\frac{\frac{\ell}{t}}{\tan k}}{\frac{k}{t}}\right)}\]

   12. Applied associate-*r*11.2

       \[\leadsto \color{blue}{\left(\frac{\frac{2}{k}}{\sin k} \cdot \frac{\frac{\frac{\ell}{t}}{1}}{1}\right) \cdot \frac{\frac{\frac{\ell}{t}}{\tan k}}{\frac{k}{t}}}\]

   13. Simplified7.1

       \[\leadsto \left(\frac{\frac{2}{k}}{\sin k} \cdot \frac{\frac{\frac{\ell}{t}}{1}}{1}\right) \cdot \color{blue}{\frac{\frac{\ell}{k}}{\tan k}}\]
   14. Using strategy rm

   15. Applied pow17.1

       \[\leadsto \left(\frac{\frac{2}{k}}{\sin k} \cdot \color{blue}{{\left(\frac{\frac{\frac{\ell}{t}}{1}}{1}\right)}^{1}}\right) \cdot \frac{\frac{\ell}{k}}{\tan k}\]

   16. Applied pow17.1

       \[\leadsto \left(\color{blue}{{\left(\frac{\frac{2}{k}}{\sin k}\right)}^{1}} \cdot {\left(\frac{\frac{\frac{\ell}{t}}{1}}{1}\right)}^{1}\right) \cdot \frac{\frac{\ell}{k}}{\tan k}\]

   17. Applied pow-prod-down7.1

       \[\leadsto \color{blue}{{\left(\frac{\frac{2}{k}}{\sin k} \cdot \frac{\frac{\frac{\ell}{t}}{1}}{1}\right)}^{1}} \cdot \frac{\frac{\ell}{k}}{\tan k}\]

   18. Simplified5.0

       \[\leadsto {\color{blue}{\left(\frac{\frac{\ell \cdot 2}{\sin k}}{t \cdot k}\right)}}^{1} \cdot \frac{\frac{\ell}{k}}{\tan k}\]
3. Recombined 3 regimes into one program.

4. Final simplification7.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \le 1.4114890122776786 \cdot 10^{+29}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{t}}{k} \cdot \frac{2}{\sin k}}{\frac{\tan k}{\frac{\ell}{k}}}\\ \mathbf{elif}\;k \le 2.3301504039808588 \cdot 10^{+240}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{k}}{\frac{k}{\ell}} \cdot \frac{2}{t}}{\tan k \cdot \sin k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{\tan k} \cdot \frac{\frac{2 \cdot \ell}{\sin k}}{t \cdot k}\\ \end{array}\]

Runtime

Time bar (total: 2.9m)Debug logProfile

herbie shell --seed 2018274 +o rules:numerics
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10-)"
  (/ 2 (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (- (+ 1 (pow (/ k t) 2)) 1))))